Math Snap

STEP 1

1. We are dealing with a chi-square distribution with 8 degrees of freedom.
2. The task is to find the area to the right of 6 under this chi-square distribution curve.
3. The area under the curve represents probabilities, and we will use statistical tables or software to find this area.
4. The result should be rounded to three decimal places.

STEP 2

1. Understand the chi-square distribution and degrees of freedom.
2. Use a chi-square distribution table or software to find the cumulative probability up to 6.
3. Calculate the area to the right of 6 by subtracting the cumulative probability from 1.

STEP 3

Understand that a chi-square distribution is a continuous probability distribution that is skewed to the right, and it is defined by its degrees of freedom. In this case, we have 8 degrees of freedom.

STEP 4

Use a chi-square distribution table or statistical software to find the cumulative probability $P(X \leq 6)$ for a chi-square distribution with 8 degrees of freedom.

STEP 5

Using a chi-square table or software, find $P(X \leq 6)$. For 8 degrees of freedom, this cumulative probability is approximately 0.3446.

Calculate the area to the right of 6, which is $1 - P(X \leq 6)$.
$$1 - 0.3446 = 0.6554$$ Round the result to three decimal places:
$$\boxed{0.655}$$ The area of the shaded region is:
$$\boxed{0.655}$$